Stejskal-Tanner equation

The diffusion experiments have been performed on a Bruker Avance 500 NMR spectrometer operating at a Larmor frequency of 500 MHz for protons using a DIFF 30 probehead generating a maximum gradient strength of 12 T/m. The gradient pulse duration 8 and diffusion time A have been adjusted between 0.7 to 1A ms for 8 and 10 to 15 ms for A respectively. The narrow distribution of the molecular weight distributiOTi permits the determination of the diffusion coefficients by a linear fit to the Stejskal-Tanner equation [23]. 

Figure 2.3 Typical experimental data from a PFG-NMR diffusion experiment with the fitted Stejskal-Tanner equation fEq. 2.5 shown as the linel.

Successive experiments varying the A, or more commonly tire gradient strength, provide the data points for a non-linear least squares fit of Equation 2.5 to solve for the diffusion coefficient. Typical experimental data with the fitted Stejskal-Tanner equation (Eq. 2.5 is shown in Figure 2.3. To produce a gradient pulse of opposite sign to the first, the polarity of the gradient coil can be physically reversed. However, this does not produce the desired level of reproducibility of the pulses, and modern NMR systems use the tt-RF pulse to cause the same effect. 

Stejskal and Tanner devised a quantitative relationship of the echo attenuation as a function of gradient strength, the Stejskal-Tanner equation [71] ... 

For self-diffusion NMR measnrements, ME samples are filled into standard NMR tubes. Tetramethylsilane dissolved in D2O and sealed into a capillary is commonly used as reference. H-NMR spectra of the pure components and the MEs are acquired and Fourier transformed. For analysis, peaks of the ME formulation can be assigned to the individnal components by comparison to spectra obtained from the pnre ME components. To obtain the data for exponential modelling using the Stejskal-Tanner equation (Equation 9.4), one or more peaks of the component are either integrated or analysis is performed on the peak height. Selfdiffusion coefficients ranging from about 10 to 10 m s" are typically observed in solutions at ambient temperature [73]. 

Van Lokeren and co-workers reported a study of polymer mixtures in solution using pulsed field-gradient spin eeho NMR. They proposed a quantification approach in which the fraetions of different components were obtained explicitly including relaxation effeets in the Stejskal-Tanner equation. 

This equation agrees with the expression originally derived by Tanner and Stejskal (the expression reproduced in the book by Callaghan misses the normalization factor I/a). 

Equation (35) is identical with the corresponding expression of Tanner and Stejskal and as reproduced in Callaghan s book. Note that expression (35) only results when the normalization constant l/a is taken into account in the propagator (34). 

In many systems, the diffusion of small molecules is not free but is restricted. Examples of systems with restricted diffusion are molecules in cellular compartments, fluids between long flat plates, lamellar and vesicular systems, water-filled pores in rocks, and small molecules in colloidal suspensions. If the time during which the molecular diffusion is monitored in the experiment (the time between rf pulses in the continuous gradient experiment and A-6/3 in a pulse gradient diffusion experiment) is much longer than the time for a molecule to travel to a boundary, then the calculated D will be too small (Woessner, 1963). In such cases of restricted diffusion, appropriate equations must be derived for the particular geometry of the constraint (Stejskal, 1965 Wayne and Cotts, 1966 Robertson, 1966 Tanner and Stejskal, 1968 Boss and Stejskal, 1968). In favorable cases, the diffusion measurement can yield not only D but an estimate of the restraining dimension as well. 

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